Let A be an invertible nn matrix, and b and x be n-vectors satisfying Ax=b. Suppose we perturb the jth entry of b by =0, so b becomes b^=b+ej. Let x^ be the n-vector that satisfies Ax^=b^, i.e., the solution of the linear equations using the perturbed right hand side. We are interested in xx^, which is how much the solution changes due to the change in the right-hand side. The ratio xx^/ gives the sensitivity of the solution to changes (perturbations) of the jth entry of b.

(a) Show that xx^ does not depend on b; it depends on the matrix A,,andj.

(b) Based on your answer to part (a), describe how you would find the index j that maximizes the value of xx^.